An almost sure functional limit theorem at zero for a class of Levy processes normed by the square root function, and applications

A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Levy processes X(t) for which lim sup(t down arrow 0) vertical bar X(t)vertical bar/root t epsilon (0, infinity) occurs almost surely (a.s.). For such processes we have a kind of almost sure "iterated logarithm" result, but without the logs. In the present paper we prove a functional version of this result, which then opens the way to various interesting applications obtained via a continuous mapping theorem. We set these out in a rigorous framework, including a characterisation of the existence of an a.s. cluster set for the interpolated process, appropriate to the continuous time situation. The applications relate to functional laws for the supremum, reflected and a variety of other processes, including a class of stochastic differential equations, where we aim to give as informative a description as we can of the functional limit sets.